Towards classification of multiple-end solutions to the Allen-Cahn equation in \(\mathbb{R}^2\).

*(English)*Zbl 1262.35011Summary: An entire solution of the Allen-Cahn equation \(\Delta u = f(u)\), where \(f\) is an odd function and has exactly three zeros at \(\pm 1\) and \(0\), e.g. \(f(u) = u(u^2 - 1)\), is called a \(2k\)-ended solution if its nodal set is asymptotic to \(2k\) half lines, and if along each of these half lines the function \(u\) looks (up to a multiplication by \(-1\)) like the one dimensional, odd, heteroclinic solution \(H\), of \(H'' = f(H)\). In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions \(U\) whose nodal lines are precisely the straight lines \(y = \pm x\). We describe the connected components of the moduli space of \(4\)-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all \(4\)-ended solutions are continuous deformations of the saddle solution.